Self Organizing Sequence

Consider the sequence with the following properties.

The first term \(a_1\) is any natural number.

The subsequent numbers follow the rule

\[a_{n+1}=\left\{\begin{array}{lr}a_n/2 & if\ a_n\ is\ even\\ 3a_n+1 & if\ a_n\ is\ odd \end{array}\right.\]

What is the value of \[\lim_{n \to \infty} \frac{\sum_{i=1}^n a_i}{n}\]

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